1-isomorphic groups

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This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Suppose $G_{1}$ and $G_{2}$ are groups. We say that $G_{1}$ and $G_{2}$ are 1-isomorphic if there exists a 1-isomorphism between $G_{1}$ and $G_{2}$ , i.e., a 1-homomorphism of groups from $G_{1}$ to $G_{2}$ whose inverse is also a 1-homomorphism. In other words, there is a bijection between $G_{1}$ and $G_{2}$ whose restriction to any cyclic subgroup on either side is an isomorphism to its image.

Finite version

Two finite groups that are 1-isomorphic are termed 1-isomorphic finite groups. There are many equivalent characterizations of 1-isomorphic finite groups.

Relation with other properties

Stronger properties

Any Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring. This shows that many groups of small prime power order are 1-isomorphic to abelian groups. Further information: Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring

Weaker properties

See 1-isomorphic finite groups#Weaker properties